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Valuations and orderings on the real Weyl algebra
Vukšić, Lara
The first Weyl algebra ▫${\cal A}_1(k)$▫ over a field ▫$k$▫ is the ▫$k$▫-algebra with two generators ▫$x, y$▫ subject to ▫$[y, x] = 1$▫ and was first introduced during the development of quantum ...mechanics. In this article, we classify all valuations on the real Weyl algebra ▫${\cal A}_1({\mathbb R})$▫ whose residue field is ▫${\mathbb R}$▫. We then use a noncommutative version of the Baer-Krull theorem to classify all orderings on ▫${\cal A}_1({\mathbb R})$▫. As a byproduct of our studies, we settle two open problems in real algebraic geometry. First, we show that not all orderings on ▫${\cal A}_1({\mathbb R})$▫ extend to an ordering on a larger ring ▫$R[y; \delta]$▫, where ▫$R$▫ is the ring of Puiseux series, introduced by Marshall and Zhang in 2000, and characterize the orderings that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky’s theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.
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